From k-essence to generalised Galileons (1103.3260v1)

Abstract: We determine the most general scalar field theories which have an action that depends on derivatives of order two or less, and have equations of motion that stay second order and lower on flat space-time. We show that those theories can all be obtained from linear combinations of Lagrangians made by multiplying a particular form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. We also obtain curved space-time extensions of those theories which have second order field equations for both the metric and the scalar field. This provide the most general extension, under the condition that field equations stay second order, of k-essence, Galileons, k-Mouflage as well as of the kinetically braided scalars. It also gives the most general action for a scalar classicalizer, which has second order field equations. We discuss the relation between our construction and the Euler hierachies of Fairlie et al, showing in particular that Euler hierachies allow one to obtain the most general theory when the latter is shift symmetric. As a simple application of our formalism, we give the covariantized version of the conformal Galileon.

• The paper generalizes scalar field theories by constructing extensive k-essence and Galileon Lagrangians that avoid higher-order derivative instabilities.
• It extends these theories into curved space-time, ensuring second-order dynamics for both scalar and metric fields critical for cosmological applications.
• It exploits Euler hierarchies to reveal shift-symmetric properties, paving the way for unified treatments of modified gravity and dark energy models.

Scalar Field Theories: From k-essence to Generalized Galileons

The paper "From k-essence to generalized Galileons" delivers a comprehensive formulation and classification of scalar field theories that maintain second-order equations of motion without propagating higher-order derivatives, thereby avoiding ghost instabilities. The authors meticulously explore the mathematical constructs linking these models, extending the reach of established theories like k-essence and Galileons to a wider array of scalar field actions.

Key Contributions

This work accomplishes several seminal objectives:

1. General Formulation of Scalar Theories: The authors derive the most extensive form of scalar field actions that are dependent on first and second derivatives, ensuring the resulting field equations do not exceed second order even in flat space-time. In essence, they demonstrate that these theories can be obtained as a linear combination of Galileon-type Lagrangians multiplied by arbitrary functions of the scalar field and its derivatives.
2. Curved Space-time Extensions: The transition from flat to curved space-time is a non-trivial step in scalar field theories, particularly regarding maintaining second-order equations for both the scalar field and the metric. The paper effectively constructs these curved space extensions, thus providing a framework to paper cosmological and astrophysical phenomena under various gravitational conditions.
3. Euler Hierarchies: Leveraging the Euler hierarchies developed by Fairlie and colleagues, the researchers expound how these hierarchies can be viewed as a means to produce shift-symmetric scalar theories. The Euler hierarchies offer an alternative viewpoint, decomposing scalar field theories into a sequence of nested field equations that naturally generate the desired properties of scalar interactions.

Implications and Future Prospects

The implications of this work are significant both theoretically and practically:

• Theoretical Impact: The results provide a unified structure encompassing diverse scalar-tensor theories within a single theoretical framework. By solidifying the mathematical underpinnings of such models, this paper paves the way for further theoretical explorations into the nature of modified gravity and dark energy.
• Practical Applications: Curved space-time generalizations allow for the simulation of scalar field dynamics in cosmology, particularly in contexts dealing with cosmic inflation and late-time accelerated expansion scenarios. With the robust mathematical foundation provided, researchers can explore new avenues in cosmological modeling and potentially reconcile discrepancies between theory and observations.

Moving forward, the expansion of these scalar theories into multi-field models and generalizations to higher derivative actions will be pivotal, especially those incorporating more complex symmetry structures or coupling mechanisms. Understanding the full spectrum of solutions within these extended Galileon frameworks could offer new insights into early-universe dynamics and dark sector physics.

In conclusion, "From k-essence to generalized Galileons" stands as a vital reference point for researchers in the theoretical physics community. The intricate mathematical analysis and broad applicability of the results promise continued relevance as the field evolves, ensuring these scalar theories remain integral to future advances in cosmology and gravitational theories.